Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 15 (2020), 399 -- 418

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CAPUTO TYPE FRACTIONAL DIFFERENTIAL EQUATION WITH NONLOCAL ERDÉLYI-KOBER TYPE INTEGRAL BOUNDARY CONDITIONS IN BANACH SPACES

Abdelatif Boutiara, Maamar Benbachir and Kaddour Guerbati

Abstract. In this paper, we study nonlocal boundary value problems of nonlinear Caputo fractional differential equations supplemented with Erdélyi-Kober type fractional integral boundary conditions. Existence results are obtained by applying the Mönch's fixed point theorem and the technique of measures of noncompactness. An example illustrating the main result is also constructed.

2020 Mathematics Subject Classification: 26A33; 34A60
Keywords: fractional differential equation; Erdélyi-Kober fractional integral conditions; Caputo fractional derivative; Kuratowski measures of noncompactness; Mönch fixed point theorems.

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References

  1. R. P. Agarwal, B. Ahmad and A. Alsaedi, Fractional-order differential equations with anti-periodic boundary conditions: a survey, Bound. Value Probl. 2017, Paper No. 173, 27 pp. MR3736605. Zbl07173754

  2. R. P. Agarwal, M. Meehan and D. O'Regan,, Donal. Fixed point theory and applications. Cambridge Tracts in Mathematics, 141. Cambridge University Press, Cambridge, 2001. x+170 pp. ISBN: 0-521-80250-4 MR1825411. Zbl1042.54507.

  3. A. Aghajani, A. M. Tehrani, D. O'Regan, Some new fixed point results via the concept of measure of noncompactness. Filomat 29 (2015), no. 6, 1209--1216. MR3359309. Zbl06749089

  4. B. Ahmad, S.K. Ntouyas, J.Tariboon, A.Alsaedi, Caputo type fractional differential equations with nonlocal Riemann-Liouville and Erdélyi-Kober type integral boundary conditions. Filomat 31 (2017), no. 14, 4515--4529. MR3730375

  5. R. R. Akhmerov, M. I. Kamenskii, A. S. Patapov, A. E. Rodkina and B. N. Sadovskii, Measures of noncompactness and condensing operators. Translated from the 1986 Russian original by A. Iacob. Operator Theory: Advances and Applications, 55. Birkhäuser Verlag, Basel, 1992. \rm viii+249 pp. ISBN: 3-7643-2716-2 MR1153247. Zbl0623.47070.

  6. J. C. Alvàrez, Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces. (Spanish) Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid 79 (1985), no. 1-2, 53--66. MR0835168

  7. J. Banas and K. Goebel, Measures of noncompactness in Banach spaces. Lecture Notes in Pure and Applied Mathematics, 60. Marcel Dekker, Inc., New York, 1980. \rm vi+97 pp. ISBN: 0-8247-1248-X MR0591679. Zbl0441.47056.

  8. J. Banas, M. Jleli, M. Mursaleen, et al. Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, Springer Nature Singapore, 2017. Zbl1371.47001. Zbl1371.47001.

  9. M. Benchohra, J. Henderson, D. Seba, Measure of noncompactness and fractional differential equations in Banach spaces. Commun. Appl. Anal. 12 (2008), no. 4, 419--427. MR2494987. Zbl1182.26007.

  10. M. Benchohra, G. M. N'Guérékata, D. Seba, Measure of noncompactness and nondensely defined semilinear functional differential equations with fractional order. Cubo 12 (2010), no. 3, 35--48. MR2779372. Zbl1219.34100.

  11. A. Boutiara, K. Guerbati, M. Benbachir, Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Mathematics, 5(1), (2019): 259-272.

  12. L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162 (1991), no. 2, 494--505. MR1137634(92m:35005). Zbl0748.34040

  13. L. Byszewski, Lakshmikantham, V. Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40 (1991), no. 1, 11--19. MR1121321(98i:34118). Zbl0776.34049.

  14. T. Chen and W. Liu, An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator. Appl. Math. Lett. 25 (2012), no. 11, 1671--1675. MR2957733. Zbl1248.35219.

  15. A. Erdélyi, H. Kober, Some remarks on Hankel transforms. Quart. J. Math. Oxford Ser. 11 (1940), 212--221. MR0003270. Zbl0025.18601.

  16. C. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 61 (2011), no. 2, 191--202. MR2754129. Zbi1211.39002.

  17. D. Guo, V. Lakshmikantham, X. Liu, Nonlinear integral equations in abstract spaces. Mathematics and its Applications, 373. Kluwer Academic Publishers Group, Dordrecht, 1996. \rm viii+341 pp. ISBN: 0-7923-4144-9 MR1418859. Zbl0866.45004.

  18. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. xvi+523 pp. ISBN: 978-0-444-51832-3; 0-444-51832-0 MR2218073(2007a:34002). Zbl1092.45003.

  19. H. Kober, On fractional integrals and derivatives. Quart. J. Math. Oxford Ser. 11 (1940), 193--211. MR0003269. Zbl0025.18502.

  20. F. Li, Gaston M. N’ Guérékata, An existence result for neutral delay integrodifferential equations with fractional order and nonlocal conditions. Abstr. Appl. Anal. 2011, Art. ID 952782, 20 pp. MR2846248. Zbl1269.45006.

  21. F. Li, J. Liang, H. K. Xu, Existence of mild solutions for fractional integrodifferential equation of Sobolev type with nonlocal conditions, J. Math. Anal. Appl., 391 (2012), 510--525.

  22. K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993. \rm xvi+366 pp. ISBN: 0-471-58884-9 MR1219954(94e:26013). Zbl0789.26002.

  23. H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4 (1980), no. 5, 985--999. MR0586861. Zbl0462.34041

  24. H. Mönch, von Harten, Gerd-Friedrich. On the Cauchy problem for ordinary differential equations in Banach spaces. Arch. Math. (Basel) 39 (1982), no. 2, 153--160. MR0675655. Zbl0496.34033.

  25. K. B. Oldham, J. Spanier, The fractional calculus. Theory and applications of differentiation and integration to arbitrary order. With an annotated chronological bibliography by Bertram Ross. Mathematics in Science and Engineering, Vol. 111. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. \rm xiii+234 pp. MR0361633. Zbl0292.26011.

  26. I. Podlubny, Fractional Differential Equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications., Academic Press, San Diego (1999). MR1926477(2001m:22005). Zbl 0924.34008.

  27. S. Szufla, On the application of measure of noncompactness to existence theorems. Rend. Sem. Mat. Univ. Padova 75 (1986), 1--14. MR0847653. Zbl0589.45007.

  28. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives. Theory and applications. Edited and with a foreword by S. M. Nikolski\i. Translated from the 1987 Russian original. Revised by the authors. Gordon and Breach Science Publishers, Yverdon, 1993. \rm xxxvi+976 pp. ISBN: 2-88124-864-0 MR1347689. Zbl0818.26003.

  29. N.Thongsalee, S.K.Ntouyas, J.Tariboon, Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdélyi-Kober fractional integral conditions. Fract. Calc. Appl. Anal. 19 (2016), no. 2, 480--497. MR3513006. Zbl1339.34015.

  30. H. E. Zhang, Nonlocal boundary value problems of fractional order at resonance with integral conditions. Adv. Difference Equ. 2017, Paper No. 326, 12 pp. MR3712505. Zbl07002465.



Abdellatif Boutiara
Laboratory of Mathematics and Applied Sciences University of Ghardaia, 47000, Algeria.
e-mail: Boutiara_a@yahoo.com

Maamar Benbachir
Faculty of Sciences, Saad Dahlab University, Blida1, Algeria.
e-mail: mbenbachir2001@gmail.com

Kaddour Guerbati
Laboratory of Mathematics and Applied Sciences University of Ghardaia, 47000, Algeria.
e-mail: guerbati_k@yahoo.com





http://www.utgjiu.ro/math/sma